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It is common for matrices in industrial applications of linear programming to have a large proportion of zero coefficients. While every item (raw material, intermediate material, end item, equipment item) in, say, a petroleum refinery may be indirectly related to every other, any particular process uses few of these. Thus the matrix describing petroleum technology has a small percentage of non-zeros. If spacial or temporal distinctions are introduced into the model the percentage of non-zeros generally falls further. The present paper discusses a form of inverse which is especially convenient to obtain and use for matrices with a high percentage of zeros. The application of this form of inverse in linear programming is also discussed.
Harry M. Markowitz (Mon,) studied this question.